WEBVTT 00:00:00.000 --> 00:00:11.340 [SQUEAKING] [RUSTLING] [CLICKING] 00:00:11.340 --> 00:00:14.290 SCOTT HUGHES: All right, so in today's recorded lecture, 00:00:14.290 --> 00:00:17.260 I would like to pick up where we started-- 00:00:17.260 --> 00:00:17.760 excuse me. 00:00:17.760 --> 00:00:20.400 I'd like to pick up where we stopped last time. 00:00:20.400 --> 00:00:24.660 So I discussed the Einstein field equations 00:00:24.660 --> 00:00:26.010 in the previous two lectures. 00:00:26.010 --> 00:00:29.520 I derived them first from the method 00:00:29.520 --> 00:00:32.910 that was used by Einstein in his original work on the subject. 00:00:32.910 --> 00:00:36.960 And then I laid out the way of coming to the Einstein field 00:00:36.960 --> 00:00:38.940 equations using an action principle, 00:00:38.940 --> 00:00:42.360 using what we call the Einstein Hilbert action. 00:00:42.360 --> 00:00:50.920 Both of them lead us to this remarkably simple equation, 00:00:50.920 --> 00:00:56.730 if you think about it in terms simply of the curvature tensor. 00:00:56.730 --> 00:01:01.860 This is saying that a particular version of the curvature. 00:01:01.860 --> 00:01:03.470 You start with the Riemann tensor. 00:01:03.470 --> 00:01:05.600 You trace over two indices. 00:01:05.600 --> 00:01:09.620 You reverse the trace such that this whole thing 00:01:09.620 --> 00:01:10.760 has zero divergence. 00:01:10.760 --> 00:01:14.630 And you simply equate that to the stress energy tensor 00:01:14.630 --> 00:01:18.380 with a coupling factor with a complex constant 00:01:18.380 --> 00:01:20.120 proportionality that ensures that this 00:01:20.120 --> 00:01:22.430 recovers the Newtonian limit. 00:01:22.430 --> 00:01:24.950 The Einstein Hilbert exercise demonstrated 00:01:24.950 --> 00:01:29.270 that this is in a very quantifiable way, the simplest 00:01:29.270 --> 00:01:32.420 possible way of developing a theory of gravity 00:01:32.420 --> 00:01:34.528 in this framework. 00:01:34.528 --> 00:01:36.070 The remainder of this course is going 00:01:36.070 --> 00:01:38.780 to be dedicated to solving this equation, 00:01:38.780 --> 00:01:41.000 and exploring the properties of the solutions that 00:01:41.000 --> 00:01:42.578 arise from this. 00:01:42.578 --> 00:01:44.120 And so let me continue the discussion 00:01:44.120 --> 00:01:48.000 I began at the end of the previous lecture. 00:01:48.000 --> 00:01:50.120 We are going to find it very useful 00:01:50.120 --> 00:02:08.740 to regard this as a set of differential equations 00:02:08.740 --> 00:02:17.090 for the spacetime metric given a source. 00:02:17.090 --> 00:02:21.080 That, after all, is how we typically 00:02:21.080 --> 00:02:24.080 think of solving for fields given a particular source. 00:02:24.080 --> 00:02:26.080 And just pardon me while I make sure this is on. 00:02:26.080 --> 00:02:26.600 It is. 00:02:29.330 --> 00:02:31.190 I give you a distribution of mass. 00:02:31.190 --> 00:02:33.350 You compute the Newtonian gravitational potential 00:02:33.350 --> 00:02:33.850 for that. 00:02:33.850 --> 00:02:36.650 I give you a distribution of currents and fields. 00:02:36.650 --> 00:02:38.720 You calculate the electric and magnetic fields 00:02:38.720 --> 00:02:40.100 that arise from that. 00:02:40.100 --> 00:02:43.220 So I give you some distribution of mass and energy. 00:02:43.220 --> 00:02:45.820 You compute the spacetime that arises from that. 00:02:45.820 --> 00:02:47.690 But let's stop before we dig into this, 00:02:47.690 --> 00:02:52.940 and look at what this actually means 00:02:52.940 --> 00:02:55.460 given the mathematical equations that we have. 00:02:55.460 --> 00:03:01.310 So G alpha beta is the Einstein tensor. 00:03:01.310 --> 00:03:05.060 I construct it by taking several derivatives of the metric. 00:03:05.060 --> 00:03:06.770 I first make my Christoffel symbols. 00:03:06.770 --> 00:03:08.900 I combine those Christoffel symbols and derivatives 00:03:08.900 --> 00:03:11.600 of the Christoffel symbols to make the Riemann tensor. 00:03:11.600 --> 00:03:13.550 I hit it with another power of the metric 00:03:13.550 --> 00:03:16.157 in order to trace and get the Ricci tensor. 00:03:16.157 --> 00:03:18.740 I combine it with the trace of the Ricci tensor and the metric 00:03:18.740 --> 00:03:20.270 to get the Einstein. 00:03:20.270 --> 00:03:24.800 Schematically, I can think of G alpha beta 00:03:24.800 --> 00:03:29.420 as some very complicated linear-- 00:03:29.420 --> 00:03:32.600 excuse me, some very complicated nonlinear differential 00:03:32.600 --> 00:03:36.640 operator acting on the metric. 00:03:55.910 --> 00:03:58.760 So thinking about this is just a differential equation 00:03:58.760 --> 00:03:59.750 for the metric. 00:03:59.750 --> 00:04:03.360 The left hand side of this equation is a bit of a mess. 00:04:03.360 --> 00:04:05.680 Unfortunately, the right hand side can be a mess too. 00:04:05.680 --> 00:04:07.680 Let's think about this for a particular example. 00:04:07.680 --> 00:04:11.430 Suppose I choose as my force-- 00:04:11.430 --> 00:04:12.780 my source-- a perfect fluid. 00:04:21.000 --> 00:04:27.653 Well, then my right hand side is going 00:04:27.653 --> 00:04:30.070 to be something that involves the density and the pressure 00:04:30.070 --> 00:04:41.060 of that fluid, the fluid velocity, and then the metric. 00:04:41.060 --> 00:04:43.460 OK, so if I'm thinking about this as a differential 00:04:43.460 --> 00:04:45.860 equation for the metric, the metric 00:04:45.860 --> 00:04:49.080 is appearing under this differential operator 00:04:49.080 --> 00:04:51.030 on the left hand side, and explicitly 00:04:51.030 --> 00:04:52.980 in the source on the right hand side. 00:04:52.980 --> 00:04:55.710 Oh, and by the way, don't forget my fluid needs 00:04:55.710 --> 00:04:56.490 to be normalized. 00:04:56.490 --> 00:04:59.040 My fluid flow velocity needs to be normalized. 00:04:59.040 --> 00:05:06.560 So I have a further constraint, that the complement to the four 00:05:06.560 --> 00:05:10.370 velocity are related to each other by the spacetime metric. 00:05:10.370 --> 00:05:14.390 So if I am going to regard this as just a differential equation 00:05:14.390 --> 00:05:19.040 for the space time metric. 00:05:19.040 --> 00:05:22.840 In general, we're in for a world of pain. 00:05:22.840 --> 00:05:27.720 So as I described at the end of the previous lecture, 00:05:27.720 --> 00:05:31.370 we are going to examine how to solve this equation. 00:05:31.370 --> 00:05:33.650 In what's left of 8.962, we're going 00:05:33.650 --> 00:05:36.850 to look at three routes to solving this thing. 00:05:36.850 --> 00:05:42.300 The one that we will begin to talk about today 00:05:42.300 --> 00:05:48.280 is we solve this for what I will define a little bit more 00:05:48.280 --> 00:05:50.125 precisely in a moment as weak gravity. 00:05:55.670 --> 00:05:57.670 And what this is going to mean is that I only 00:05:57.670 --> 00:06:05.320 consider space times that are in a way that can be quantified 00:06:05.320 --> 00:06:11.480 close to flat space time. 00:06:11.480 --> 00:06:15.350 Method two will be to consider symmetric solutions. 00:06:23.460 --> 00:06:27.210 Part of the reason the general framework is so complicated 00:06:27.210 --> 00:06:30.330 is that there are in general, 10 of these 00:06:30.330 --> 00:06:32.850 coupled non-linear differential equations. 00:06:32.850 --> 00:06:34.530 When we introduce symmetries, or we 00:06:34.530 --> 00:06:37.950 consider things like static solutions 00:06:37.950 --> 00:06:41.730 or stationary solutions that don't have any time dynamics. 00:06:41.730 --> 00:06:43.742 That at least reduces the number of equations 00:06:43.742 --> 00:06:44.700 we need to worry about. 00:06:44.700 --> 00:06:49.590 They may still be coupled non-linear and complicated, 00:06:49.590 --> 00:06:51.428 but hopefully, maybe we can reduce 00:06:51.428 --> 00:06:52.970 those from 10 of these things we need 00:06:52.970 --> 00:06:56.280 to worry about to just a small number them, one, or two, 00:06:56.280 --> 00:06:58.350 or three. 00:06:58.350 --> 00:07:00.390 Makes it at least a little easier. 00:07:00.390 --> 00:07:05.340 In truth, symmetric solutions-- 00:07:05.340 --> 00:07:13.310 if you can then add techniques for perturbing around them, 00:07:13.310 --> 00:07:15.280 these turn out to be tremendously powerful. 00:07:20.910 --> 00:07:24.210 My own research career is-- 00:07:24.210 --> 00:07:26.370 uses this technique a tremendous amount. 00:07:28.960 --> 00:07:33.713 Finally, just basically say, you know what? 00:07:33.713 --> 00:07:35.380 Let's just dive in and solve this puppy. 00:07:41.150 --> 00:07:46.900 Just do a numerical solution of the whole monster, 00:07:46.900 --> 00:07:48.482 no simplifications. 00:07:55.100 --> 00:07:59.030 Eminent members of our field have dedicated entire careers 00:07:59.030 --> 00:08:00.920 to item three. 00:08:00.920 --> 00:08:03.260 I will give you an introduction to the main concepts 00:08:03.260 --> 00:08:05.660 in the last lecture. 00:08:05.660 --> 00:08:07.240 But it's not something we're going 00:08:07.240 --> 00:08:11.210 to be able to explore in much detail in this class. 00:08:11.210 --> 00:08:13.403 We're going to do one in a fair amount of detail. 00:08:13.403 --> 00:08:15.320 We will look at a couple of the most important 00:08:15.320 --> 00:08:16.820 symmetric solutions, so that you can 00:08:16.820 --> 00:08:21.315 see how these techniques work. 00:08:21.315 --> 00:08:22.690 And then in my last two lectures, 00:08:22.690 --> 00:08:24.020 I'm going to describe a little bit about what 00:08:24.020 --> 00:08:26.252 happens when you perturb some of the most interesting 00:08:26.252 --> 00:08:27.085 symmetric solutions. 00:08:27.085 --> 00:08:33.970 And we'll talk about numerical solutions for the general case. 00:08:33.970 --> 00:08:35.850 All right, so let's begin. 00:08:35.850 --> 00:08:37.370 We'll begin with choice one. 00:08:37.370 --> 00:08:44.900 Look at weak gravity, also known as 00:08:44.900 --> 00:08:47.156 linearized general relativity. 00:08:54.460 --> 00:08:57.550 So linearized general relativity is a situation 00:08:57.550 --> 00:09:00.100 in which we are only going to consider space times that 00:09:00.100 --> 00:09:01.360 are nearly flat. 00:09:15.650 --> 00:09:28.250 If I am in this situation, then I can choose coordinates, 00:09:28.250 --> 00:09:37.890 such that my space time metric is the metric of flat space 00:09:37.890 --> 00:09:49.120 time plus a tensor H-alpha beta, all of whose components-- 00:09:52.338 --> 00:09:54.380 so this notation that I'm sort of inventing here, 00:09:54.380 --> 00:09:57.320 double bars around H-alpha beta. 00:09:57.320 --> 00:10:07.830 This means the magnitude of H-alpha beta's components. 00:10:13.790 --> 00:10:18.770 These all must be much, much less than one. 00:10:18.770 --> 00:10:22.190 When you are in such a coordinate system 00:10:22.190 --> 00:10:28.070 you are in what we call nearly Lorenz coordinates. 00:10:39.040 --> 00:10:41.110 Such a coordinate system is as close 00:10:41.110 --> 00:10:43.510 to a globally inertial coordinate system 00:10:43.510 --> 00:10:45.610 as is possible to make. 00:10:45.610 --> 00:10:47.660 There are other coordinate choices we could make. 00:10:47.660 --> 00:10:50.050 So for instance, you're working in a system like that. 00:10:50.050 --> 00:10:52.000 This basically boils down to coordinates 00:10:52.000 --> 00:10:56.320 that are Cartesian like on their spatial slices. 00:10:56.320 --> 00:10:58.240 You could work in other ones. 00:10:58.240 --> 00:10:59.740 These are particularly convenient. 00:10:59.740 --> 00:11:02.680 Because for instance, if I work in a coordinate system whose 00:11:02.680 --> 00:11:05.145 spatial sector is spherical like, 00:11:05.145 --> 00:11:07.270 well, then there's going to be some components that 00:11:07.270 --> 00:11:12.940 grow very large as I go to large radius away from some-- 00:11:12.940 --> 00:11:15.280 the source of my gravitation. 00:11:15.280 --> 00:11:18.730 And this just makes my analysis quite convenient. 00:11:18.730 --> 00:11:22.180 In particular, where I'm going to take advantage of this. 00:11:28.030 --> 00:11:30.400 Whenever I come across a term that 00:11:30.400 --> 00:11:38.420 involves the perturbations squared or to a higher power, 00:11:38.420 --> 00:11:40.040 I'm just going approximate it as zero. 00:11:40.040 --> 00:11:43.850 I will always neglect terms beyond linear, 00:11:43.850 --> 00:11:53.510 hence the term linearized GR, in my analysis. 00:12:06.980 --> 00:12:08.630 Now, there are a couple of properties, 00:12:08.630 --> 00:12:14.030 before I get into how to develop weak gravity, linearized GR. 00:12:14.030 --> 00:12:17.390 I want to discuss a little bit some of the properties 00:12:17.390 --> 00:12:22.100 of spacetimes of this form. 00:12:22.100 --> 00:12:25.710 What are the particularly important properties of these? 00:12:25.710 --> 00:12:28.269 So let's consider coordinate transformations. 00:12:50.730 --> 00:12:54.870 My space time metric is a tensor, like any other. 00:12:54.870 --> 00:13:00.300 And so the usual rules pertain here, that I can change my 00:13:00.300 --> 00:13:09.360 coordinates using some matrix that 00:13:09.360 --> 00:13:11.670 relates my original coordinate system, which 00:13:11.670 --> 00:13:17.770 I've denoted without bars, to some new coordinate system that 00:13:17.770 --> 00:13:18.280 is barred. 00:13:23.780 --> 00:13:29.400 Now, recall that when we worked in flat space time, 00:13:29.400 --> 00:13:32.190 there was one category of coordinate transformations 00:13:32.190 --> 00:13:33.540 that was special. 00:13:33.540 --> 00:13:35.834 Those were the Lorentz transformations. 00:14:10.920 --> 00:14:15.110 So we are not working in flat space time. 00:14:15.110 --> 00:14:18.640 So on the face of it, we don't expect Lorentz transformations 00:14:18.640 --> 00:14:20.140 to play a particularly special role, 00:14:20.140 --> 00:14:23.260 except perhaps in the domain of a freely falling frame. 00:14:27.130 --> 00:14:28.210 But you know what? 00:14:28.210 --> 00:14:31.310 This is a nearly flat space time, so just for giggles, 00:14:31.310 --> 00:14:34.210 let's see what happens if you apply the Lorentz 00:14:34.210 --> 00:14:38.716 transformation to your nearly flat spacetime. 00:14:53.620 --> 00:15:05.510 So if I look at G mu bar nu bar as being 00:15:05.510 --> 00:15:20.740 a Lorentz transformation applied to my nearly flat metric. 00:15:20.740 --> 00:15:39.493 Well, what I get [INAUDIBLE] side is this. 00:15:39.493 --> 00:15:41.910 Now, one of the reasons why the Lorentz transformation was 00:15:41.910 --> 00:15:44.820 special in flat spacetime is that it 00:15:44.820 --> 00:15:49.740 leaves the metric of flat spacetime unchanged. 00:15:49.740 --> 00:15:57.480 And so this just maps to eta mu nu. 00:16:01.780 --> 00:16:05.850 I'm going to define what comes out here as h mu nu. 00:16:11.600 --> 00:16:14.030 I'm doing this in a fairly, I hope, obvious way. 00:16:23.190 --> 00:16:24.860 This is interesting. 00:16:24.860 --> 00:16:31.040 What this has told me is that when I apply the Lorentz 00:16:31.040 --> 00:16:35.600 transformation to my nearly flat space time, 00:16:35.600 --> 00:16:39.330 the background is unchanged. 00:16:39.330 --> 00:16:41.750 And the perturbation to the background 00:16:41.750 --> 00:16:45.260 transforms just like any tensor field 00:16:45.260 --> 00:16:47.900 would transform in flat space. 00:16:53.970 --> 00:16:57.270 Now, it should be emphasized, we are not working in flat space. 00:17:00.530 --> 00:17:03.620 We can compute curvature tensors. 00:17:03.620 --> 00:17:05.869 If we parallel transport-- 00:17:05.869 --> 00:17:09.410 If we consider two geodesics moving through the space time, 00:17:09.410 --> 00:17:12.150 we will see that parallel transport along those two 00:17:12.150 --> 00:17:14.150 geodesics, if they start out initially parallel, 00:17:14.150 --> 00:17:15.900 they do not remain parallel. 00:17:15.900 --> 00:17:18.630 So this is not flat space time. 00:17:18.630 --> 00:17:22.040 But in many ways, it's close enough 00:17:22.040 --> 00:17:25.579 that we can borrow many of the mathematical tools that 00:17:25.579 --> 00:17:28.220 were used in flat space time. 00:17:28.220 --> 00:17:31.910 In particular, we can introduce the following. 00:17:31.910 --> 00:17:39.990 Think of it as a useful fiction, which 00:17:39.990 --> 00:17:48.200 is that in this framework, we can regard 00:17:48.200 --> 00:17:51.260 the perturbation that pushes us away from flat 00:17:51.260 --> 00:18:04.090 spacetime as just an ordinary tensor field 00:18:04.090 --> 00:18:06.700 living in the manifold of special relativity, 00:18:06.700 --> 00:18:09.160 living in the eta alpha beta metric. 00:18:12.380 --> 00:18:16.990 It's worth bearing in mind in a fundamental sense, it's not. 00:18:16.990 --> 00:18:18.910 Space time is curved. 00:18:18.910 --> 00:18:22.240 h alpha beta is telling me about that curvature. 00:18:22.240 --> 00:18:24.700 But the mathematics works in such a way 00:18:24.700 --> 00:18:26.380 that you can borrow a lot of tools 00:18:26.380 --> 00:18:27.880 that we used in special relativity. 00:18:27.880 --> 00:18:30.490 And just imagine h as a tensor field 00:18:30.490 --> 00:18:33.760 living in that special relativity manifold. 00:18:39.530 --> 00:18:43.430 OK, so that's useful fact one that we 00:18:43.430 --> 00:18:51.470 want to bear in mind as we work in this nearly flat space time. 00:18:51.470 --> 00:18:57.600 Useful fact two is we want to think 00:18:57.600 --> 00:19:00.980 about what happens when we raise and lower indices. 00:19:15.260 --> 00:19:17.630 So suppose now that I'm going to regard 00:19:17.630 --> 00:19:19.940 h alpha beta as an ordinary tensor 00:19:19.940 --> 00:19:22.040 field living in this thing. 00:19:22.040 --> 00:19:24.470 I might want to know what it looks 00:19:24.470 --> 00:19:27.145 like with its indices raised. 00:19:27.145 --> 00:19:28.520 So I'm going to do what I usually 00:19:28.520 --> 00:19:33.478 do when I want to raise the indices on a tensor. 00:19:33.478 --> 00:19:34.520 I hit it with the metric. 00:19:38.190 --> 00:19:42.480 We're going to talk about what my upstairs metric looks 00:19:42.480 --> 00:19:43.590 like in just a moment. 00:19:43.590 --> 00:19:45.360 But clearly, it's going to be something 00:19:45.360 --> 00:19:49.440 that looks like the metric of flat space time plus terms 00:19:49.440 --> 00:19:51.600 that are on the order of h. 00:19:51.600 --> 00:19:57.420 Because I always drop terms of order h squared and higher, 00:19:57.420 --> 00:20:03.210 I can immediately say that this must simply 00:20:03.210 --> 00:20:10.410 be the metric of flat space time with the indices 00:20:10.410 --> 00:20:13.260 in the upstairs position acting on h. 00:20:13.260 --> 00:20:20.270 In other words, at least when I am 00:20:20.270 --> 00:20:24.830 acting on tensors that are built from the space time 00:20:24.830 --> 00:20:40.440 metric itself, I'm going to always want 00:20:40.440 --> 00:20:56.550 to raise and lower them using my flat space time, eta alpha 00:20:56.550 --> 00:20:57.870 beta. 00:20:57.870 --> 00:21:01.780 Bearing this in mind, let's carefully 00:21:01.780 --> 00:21:04.750 think about what the metric inverse actually looks like. 00:21:11.050 --> 00:21:15.550 I actually used this in one of my calculations 00:21:15.550 --> 00:21:16.820 in the previous lecture. 00:21:16.820 --> 00:21:19.935 And I said, I'm going to justify this in the next lecture. 00:21:19.935 --> 00:21:20.560 So here we are. 00:21:20.560 --> 00:21:22.850 Now we're going to justify it. 00:21:22.850 --> 00:21:25.346 So let's use this definition. 00:21:32.800 --> 00:21:35.700 So the upstairs metric is defined such 00:21:35.700 --> 00:21:38.160 that when it contracts with the downstairs metric, 00:21:38.160 --> 00:21:39.810 I get the identity back. 00:21:39.810 --> 00:21:46.590 This is the definition of the metric inverse. 00:21:46.590 --> 00:21:49.320 Working in linear theory, I know that this thing 00:21:49.320 --> 00:21:53.910 is going to be something like eta alpha beta 00:21:53.910 --> 00:21:57.450 plus a term of order h. 00:21:57.450 --> 00:21:59.700 I don't know what that term is yet, so let me just 00:21:59.700 --> 00:22:00.990 give it a new name. 00:22:00.990 --> 00:22:06.406 I'm going to call it m alpha beta. 00:22:06.406 --> 00:22:06.906 Whoops. 00:22:20.288 --> 00:22:21.830 Hopefully, the math will soon show me 00:22:21.830 --> 00:22:24.110 what this m actually is. 00:22:24.110 --> 00:22:27.335 It will be of order h, but as of yet, unknown. 00:22:56.500 --> 00:22:57.432 OK, so you know what? 00:22:57.432 --> 00:22:58.640 Let's rewrite that over here. 00:23:11.990 --> 00:23:14.320 So let's now multiply this guy out. 00:23:14.320 --> 00:23:18.340 Eta alpha beta hitting eta beta gamma. 00:23:18.340 --> 00:23:21.860 That gives me delta alpha gamma. 00:23:21.860 --> 00:23:23.770 m hitting eta. 00:23:23.770 --> 00:23:25.660 Now, remember what I just said over here. 00:23:28.360 --> 00:23:30.160 When I'm working with spacetime tensors, 00:23:30.160 --> 00:23:34.240 I always raise and lower indices using eta alpha beta. 00:23:34.240 --> 00:23:38.500 So when m alpha beta hits eta beta gamma, 00:23:38.500 --> 00:23:43.600 this is going to give me m alpha gamma. 00:23:43.600 --> 00:23:47.860 This eta hits that h. 00:23:47.860 --> 00:23:53.520 I get h, alpha upstairs, gamma downstairs. 00:23:53.520 --> 00:23:57.120 And then this guy is of order h, that guy is of order h. 00:23:57.120 --> 00:24:00.940 So additional terms of order h squared. 00:24:04.290 --> 00:24:06.860 So these guys cancel. 00:24:06.860 --> 00:24:11.690 And what I am left with is m alpha gamma 00:24:11.690 --> 00:24:16.940 equals negative h alpha gamma. 00:24:16.940 --> 00:24:20.400 I can raise my two indices, raise the gammas on both sides 00:24:20.400 --> 00:24:20.900 here. 00:24:20.900 --> 00:24:26.060 And we deduce from this that my inverse metric 00:24:26.060 --> 00:24:28.220 is eta alpha beta in the upstairs 00:24:28.220 --> 00:24:33.290 position, minus h alpha beta. 00:24:33.290 --> 00:24:34.940 At least two linear order in h. 00:24:45.560 --> 00:24:47.490 I'll just comment that what this is-- 00:24:47.490 --> 00:24:50.730 essentially, the matrix or tensor equivalent 00:24:50.730 --> 00:24:53.130 of a binomial expansion. 00:24:53.130 --> 00:24:57.450 1 over 1 plus epsilon is approximately 1 minus epsilon. 00:24:57.450 --> 00:24:59.390 That's all this is. 00:24:59.390 --> 00:25:01.380 But this is important to do. 00:25:01.380 --> 00:25:05.940 In fact, I will just sort of remark somewhat anecdotally 00:25:05.940 --> 00:25:08.640 that when I work with graduate students on projects, 00:25:08.640 --> 00:25:10.710 where we have to do things in linearized theory, 00:25:10.710 --> 00:25:13.410 getting a sign wrong here is one of the most common mistakes 00:25:13.410 --> 00:25:16.100 that people make. 00:25:16.100 --> 00:25:18.390 All right, one final detail. 00:25:45.440 --> 00:25:49.170 So this detail, as I just labeled it, 00:25:49.170 --> 00:25:51.100 is a particularly important one. 00:25:51.100 --> 00:25:55.815 We talked about general coordinate transformations. 00:25:55.815 --> 00:25:57.790 And I immediately went in to discuss a Lorentz 00:25:57.790 --> 00:25:59.500 transformation. 00:25:59.500 --> 00:26:03.040 There's a different category of coordinate transformation 00:26:03.040 --> 00:26:10.270 that plays a very important role in understanding 00:26:10.270 --> 00:26:13.893 the physics of systems that we analyze in linearized theory. 00:26:13.893 --> 00:26:15.310 So let's consider a different kind 00:26:15.310 --> 00:26:16.900 of coordinate transformation. 00:26:16.900 --> 00:26:27.790 Let's consider a coordinate shift, 00:26:27.790 --> 00:26:36.130 which I'm going to define by x alpha prime. 00:26:36.130 --> 00:26:37.930 Really, they should be x prime alpha, 00:26:37.930 --> 00:26:41.080 but you'll see why I need to name it this way right now. 00:26:41.080 --> 00:26:47.330 So this is my original coordinate, x alpha, 00:26:47.330 --> 00:26:52.490 plus some little offset that's a function of the coordinates x 00:26:52.490 --> 00:26:53.240 beta. 00:26:53.240 --> 00:26:57.740 So think of this as suppose I have a coordinate grid that 00:26:57.740 --> 00:26:58.550 looks like this. 00:27:04.550 --> 00:27:07.340 And I have some function that says, 00:27:07.340 --> 00:27:09.830 I want to consider a different system of coordinates that 00:27:09.830 --> 00:27:14.150 maybe pushes me a little bit along away from each of these 00:27:14.150 --> 00:27:16.848 coordinate lines in a way that varies 00:27:16.848 --> 00:27:18.140 as a function of position here. 00:27:21.960 --> 00:27:27.230 This notation is kind of an abuse of the indices. 00:27:27.230 --> 00:27:29.820 I am really not trying to define a coordinate invariant 00:27:29.820 --> 00:27:30.720 relationship here. 00:27:30.720 --> 00:27:33.360 I am just trying to connect two quantities, 00:27:33.360 --> 00:27:35.670 and I'm trying to connect quantities in two 00:27:35.670 --> 00:27:37.080 specific coordinate systems. 00:27:37.080 --> 00:27:41.220 And as we'll see, even though this is a bit ugly. 00:27:41.220 --> 00:27:44.005 It works well for what we want to do. 00:27:44.005 --> 00:27:45.630 So my coordinate transformation matrix. 00:28:03.060 --> 00:28:06.760 OK, so I just take the matrix of-- 00:28:06.760 --> 00:28:08.410 I developed the Jacobian-- my matrix 00:28:08.410 --> 00:28:11.410 of partial derivatives-- of the new coordinate 00:28:11.410 --> 00:28:14.350 with respect to the old coordinate in the usual way. 00:28:14.350 --> 00:28:18.530 This is going to be my first term. 00:28:18.530 --> 00:28:20.578 It's just a Kronecker delta. 00:28:20.578 --> 00:28:22.120 And then I'm going to get a term that 00:28:22.120 --> 00:28:28.450 looks like matrix of derivatives of the function that 00:28:28.450 --> 00:28:31.692 defines my infinitesimal-- defines my shift. 00:28:31.692 --> 00:28:33.400 I just gave away what I was about to say. 00:28:33.400 --> 00:28:36.190 I'm going to require [in?] my work in these nearly Lorentz 00:28:36.190 --> 00:28:39.190 coordinates, all of these entries need to be small. 00:28:44.000 --> 00:28:46.750 These will all be much, much less than 1. 00:28:46.750 --> 00:28:48.520 And so we call this an infinitesimal 00:28:48.520 --> 00:28:50.635 coordinate transformation. 00:29:08.980 --> 00:29:11.980 We are going to need to use the inverse of this guy. 00:29:24.570 --> 00:29:27.000 And using the definition of the inverse of this, 00:29:27.000 --> 00:29:30.580 saying essentially that when I take this, and I contract-- 00:29:30.580 --> 00:29:31.680 Let me put it this way. 00:29:31.680 --> 00:29:33.450 I'll just write it out. 00:29:33.450 --> 00:29:40.210 So if I compute this. 00:29:40.210 --> 00:29:43.860 I get my Kronecker delta back. 00:29:43.860 --> 00:29:47.250 Taking advantage of the smallness 00:29:47.250 --> 00:29:50.860 of the transformation. 00:29:50.860 --> 00:30:07.560 It's not terribly hard to demonstrate that 00:30:07.560 --> 00:30:09.570 what comes out of it is this. 00:30:18.810 --> 00:30:21.720 The minus sign is the key thing which I want to emphasize here. 00:30:21.720 --> 00:30:24.420 That minus sign is very similar to the minus sign 00:30:24.420 --> 00:30:25.218 that I have here. 00:30:25.218 --> 00:30:26.760 What we're doing is, again, just kind 00:30:26.760 --> 00:30:30.870 of the matrix equivalent of expanding 1 over 1 00:30:30.870 --> 00:30:32.490 plus epsilon for small epsilon. 00:30:35.880 --> 00:30:38.040 The reason that I am doing this is 00:30:38.040 --> 00:30:40.920 that I would now like to look at how 00:30:40.920 --> 00:30:45.458 the metric changes under this coordinate transformation. 00:30:55.720 --> 00:31:00.910 So what I'm going to do is define g mu nu 00:31:00.910 --> 00:31:02.230 in the new coordinate system. 00:31:11.650 --> 00:31:14.200 Usual operation. 00:31:14.200 --> 00:31:18.910 Let's now insert the many different definitions 00:31:18.910 --> 00:31:21.910 that we have introduced here. 00:31:21.910 --> 00:31:27.280 Notice that what I am using for my transformation matrix 00:31:27.280 --> 00:31:29.500 there is the inverse that I just wrote down. 00:31:45.290 --> 00:31:48.630 So let's fill that in. 00:31:48.630 --> 00:31:51.990 So I'm going to get a term that involves 00:31:51.990 --> 00:31:59.420 a Kronecker minus a matrix of partial derivatives. 00:31:59.420 --> 00:32:07.070 My other one gives me a nether Kronecker matrix 00:32:07.070 --> 00:32:09.230 of partial derivatives. 00:32:09.230 --> 00:32:13.400 And then finally, don't forget we 00:32:13.400 --> 00:32:17.150 are working in this nearly flat space time metric. 00:32:21.010 --> 00:32:23.260 And so I insert in my last term, eta alpha 00:32:23.260 --> 00:32:26.300 beta plus h alpha beta. 00:32:26.300 --> 00:32:30.620 So now, let's go and expand all of these terms out. 00:32:30.620 --> 00:32:32.440 My Kronecker, first, I get a term 00:32:32.440 --> 00:32:40.510 where both of the Kroneckers hit the metric of flat space time. 00:32:40.510 --> 00:32:42.810 So what I get is eta mu nu. 00:32:42.810 --> 00:32:45.240 Then I get a term which both the Kroneckers hit, 00:32:45.240 --> 00:32:48.400 the perturbation h alpha beta. 00:32:48.400 --> 00:32:50.740 Gives me h mu nu. 00:32:50.740 --> 00:32:52.720 Then, I'm going to get terms that 00:32:52.720 --> 00:32:59.200 involve these matrices of partial derivatives hitting 00:32:59.200 --> 00:33:00.450 the metric of flat space time. 00:33:02.980 --> 00:33:07.390 And what that's going to do is in keeping with our principle 00:33:07.390 --> 00:33:10.360 that when we're dealing with spacetime quantities, we raise 00:33:10.360 --> 00:33:14.200 and lower indices with eta. 00:33:14.200 --> 00:33:17.415 This is going to now give me-- 00:33:17.415 --> 00:33:18.540 pardon me just one moment-- 00:33:26.450 --> 00:33:28.863 a term that looks like partial-- 00:33:28.863 --> 00:33:30.530 everything in the downstairs position, d 00:33:30.530 --> 00:33:35.090 mu xi nu minus d nu xi mu. 00:33:35.090 --> 00:33:39.920 And then all the other terms are on the order 00:33:39.920 --> 00:33:47.000 of h times derivatives of the generators 00:33:47.000 --> 00:33:50.190 of my coordinate transmission. 00:33:50.190 --> 00:33:51.510 Small times small. 00:33:51.510 --> 00:33:52.870 These are infinitesimal squared. 00:33:52.870 --> 00:33:56.300 We are going to neglect them. 00:33:56.300 --> 00:33:58.970 Suppose that I insist that I have gone from one nearly 00:33:58.970 --> 00:34:01.127 flat spacetime to another. 00:34:01.127 --> 00:34:02.210 Bear in mind this picture. 00:34:02.210 --> 00:34:04.252 I'm just changing my representation a little bit. 00:34:04.252 --> 00:34:05.810 I've not changed the physics. 00:34:05.810 --> 00:34:13.639 So if I write this as eta mu prime nu prime, 00:34:13.639 --> 00:34:16.130 plus h mu prime nu prime. 00:34:16.130 --> 00:34:17.330 Well, I've got etas on both. 00:34:23.960 --> 00:34:25.480 The thing which is interesting is 00:34:25.480 --> 00:34:30.370 that I have generated a shift to my perturbation to the metric. 00:34:50.761 --> 00:34:52.219 Let's drop the primes for a second. 00:34:52.219 --> 00:34:54.420 And I'll just say that my nu, h mu 00:34:54.420 --> 00:35:09.980 nu is the old h mu nu minus the symmetrized combination 00:35:09.980 --> 00:35:12.560 of derivatives of-- 00:35:12.560 --> 00:35:16.430 the symmetrized combination of derivatives of infinitesimal 00:35:16.430 --> 00:35:17.813 coordinate transformation. 00:35:22.650 --> 00:35:25.260 Does this remind us of anything? 00:35:25.260 --> 00:35:33.870 This is starkly reminiscent of the way 00:35:33.870 --> 00:35:38.110 in which when we work with electromagnetic fields, 00:35:38.110 --> 00:35:48.480 I can take a potential, and shift it 00:35:48.480 --> 00:35:56.410 by the gradient of some scalar to generate a new potential. 00:35:56.410 --> 00:35:59.140 In so doing, what we find is that this 00:35:59.140 --> 00:36:03.370 leaves the fields unchanged. 00:36:03.370 --> 00:36:07.480 If you compute your Faraday tensor associated with this, 00:36:07.480 --> 00:36:09.320 it is unchanged. 00:36:12.250 --> 00:36:13.870 Similarly, we're going to write out 00:36:13.870 --> 00:36:16.240 the details of this in just a moment. 00:36:16.240 --> 00:36:31.840 When I generate the Riemann curvature from this, 00:36:31.840 --> 00:36:35.230 we find that although the metric has been tweaked a little bit 00:36:35.230 --> 00:36:37.210 by this coordinate transformation, 00:36:37.210 --> 00:36:39.370 Riemann is left unchanged. 00:36:55.970 --> 00:37:00.990 In acknowledgment of this, we call an infinitesimal 00:37:00.990 --> 00:37:04.020 coordinate transformation of this kind a gauge 00:37:04.020 --> 00:37:05.235 transformation. 00:37:40.440 --> 00:37:42.210 What the gauge transformation does 00:37:42.210 --> 00:37:44.910 is it allows us to change the metric, 00:37:44.910 --> 00:37:47.742 or change the way that we are representing our metric. 00:37:47.742 --> 00:37:49.200 And it's going to turn out to leave 00:37:49.200 --> 00:37:52.710 curvature tensors unchanged, in the same way that 00:37:52.710 --> 00:37:55.380 changing the potential and electrodynamics 00:37:55.380 --> 00:37:58.728 with a gauge transformation leaves our fields unchanged. 00:37:58.728 --> 00:38:01.020 And we're going to exploit this in exactly the same way 00:38:01.020 --> 00:38:02.700 that we exploit this in electrodynamics. 00:38:02.700 --> 00:38:05.490 We use this in electrodynamics in order 00:38:05.490 --> 00:38:08.790 to recast the equations governing our potentials 00:38:08.790 --> 00:38:11.640 into a form that is maximally convenient for whatever 00:38:11.640 --> 00:38:13.098 calculation we are doing right now. 00:38:13.098 --> 00:38:15.473 We're going to find-- and then we're going to derive this 00:38:15.473 --> 00:38:17.070 probably in about 20 minutes-- 00:38:17.070 --> 00:38:21.430 that the equations that govern h mu nu. 00:38:21.430 --> 00:38:23.210 If we leave things as general as possible, 00:38:23.210 --> 00:38:24.210 they're a bit of a mess. 00:38:24.210 --> 00:38:27.420 But by choosing the right gauge, we 00:38:27.420 --> 00:38:29.700 can simplify them, and wind up with a set 00:38:29.700 --> 00:38:32.640 of equations that are-- 00:38:32.640 --> 00:38:35.490 they cover all physical situations that matter, 00:38:35.490 --> 00:38:39.480 and that allow us to just cast things 00:38:39.480 --> 00:38:44.040 into a form that is much better for us to work with. 00:38:44.040 --> 00:38:45.420 All right. 00:38:45.420 --> 00:38:50.130 So we have now developed all of the sort of linguistics 00:38:50.130 --> 00:38:56.640 of linearized geometry that I want to use. 00:38:56.640 --> 00:38:58.560 Let's now go from linearized geometry 00:38:58.560 --> 00:39:01.410 to linearized gravity by running this through, and making 00:39:01.410 --> 00:39:02.740 some physics. 00:39:02.740 --> 00:39:05.730 What I want to do is look at the field 00:39:05.730 --> 00:39:08.033 equations in this framework. 00:39:11.350 --> 00:39:14.260 I am not going to run through every step of the next couple 00:39:14.260 --> 00:39:17.260 calculations. 00:39:17.260 --> 00:39:21.070 Doing so is a good illustration of the kind of calculation 00:39:21.070 --> 00:39:23.680 that a physicist likes to call straightforward but tedious. 00:39:23.680 --> 00:39:26.170 So I'm going to just write down what the results turned out 00:39:26.170 --> 00:39:27.590 to be. 00:39:27.590 --> 00:39:33.220 So let's run the metric through the machinery 00:39:33.220 --> 00:39:43.570 that we need to make all of our curvature tensors. 00:39:55.030 --> 00:39:58.350 OK, I'll remind you when we do this, we are linearizing. 00:39:58.350 --> 00:40:01.470 So anytime we see a term that looks like h squared, it dies. 00:40:01.470 --> 00:40:05.020 So we're only keeping things to linear order in h. 00:40:05.020 --> 00:40:14.270 So the first thing we find is the Riemann tensor turns 00:40:14.270 --> 00:40:22.280 into the following combination of partial derivatives 00:40:22.280 --> 00:40:23.840 of the metric perturbation h. 00:40:43.220 --> 00:40:47.300 In my notes, I have written out what 00:40:47.300 --> 00:40:54.230 happens when you switch from some original tensor h 00:40:54.230 --> 00:40:58.340 to a modified one using this gauge transformation. 00:40:58.340 --> 00:41:01.710 And what I show is that-- 00:41:01.710 --> 00:41:02.690 just a quick aside-- 00:41:08.520 --> 00:41:13.560 the gauge transformation generates 00:41:13.560 --> 00:41:21.772 a delta Riemann that looks like it's 00:41:21.772 --> 00:41:22.980 a whole bunch of-- let's see. 00:41:22.980 --> 00:41:23.813 Let's count them up. 00:41:23.813 --> 00:41:25.470 1, 2, 3, 4, 5, 6, 7, 8. 00:41:25.470 --> 00:41:26.785 You have eight terms. 00:41:26.785 --> 00:41:29.160 Of course there's eight, because there's four terms here, 00:41:29.160 --> 00:41:30.632 and you get two more for each one. 00:41:30.632 --> 00:41:33.090 So you're going to wind up with eight additional terms that 00:41:33.090 --> 00:41:35.610 involve three partial derivatives of the gauge 00:41:35.610 --> 00:41:36.547 generator. 00:41:48.740 --> 00:41:52.460 So they're of the form d cubed on xi. 00:41:52.460 --> 00:41:53.630 And it's not hard to show. 00:41:53.630 --> 00:41:54.920 You just sort of look at them. 00:41:54.920 --> 00:41:56.225 They cancel in pairs. 00:42:05.870 --> 00:42:08.970 And so delta Riemann is zero. 00:42:08.970 --> 00:42:13.352 The Riemann tensor is invariant to the gauge transformation. 00:42:17.420 --> 00:42:20.090 All right, we want to take this Riemann 00:42:20.090 --> 00:42:21.890 and use it to build the Einstein tensor. 00:42:21.890 --> 00:42:26.840 Our goal here is to make the field equation 00:42:26.840 --> 00:42:28.406 in linearized coordinates. 00:42:42.020 --> 00:42:47.540 So let's start by making the Ricci tensor. 00:42:47.540 --> 00:42:49.010 So we're going to raise and lower 00:42:49.010 --> 00:42:57.010 indices in linearized theory with the flat spacetime. 00:42:57.010 --> 00:43:07.690 So when we make this guy, what we get is this. 00:43:26.840 --> 00:43:29.960 I've introduced a couple of definitions here. 00:43:29.960 --> 00:43:31.940 One of them, you've seen before. 00:43:31.940 --> 00:43:36.980 The box operator is just a flat spacetime wave operator. 00:43:36.980 --> 00:43:42.900 And h with no indices is what I get 00:43:42.900 --> 00:43:50.885 when I trace over h using the flat background spacetime. 00:43:53.773 --> 00:43:54.690 And let's do one more. 00:44:01.520 --> 00:44:08.160 Evaluating r, I get one further contraction. 00:44:08.160 --> 00:44:17.140 And this turns out to be d alpha, d mu, h alpha mu 00:44:17.140 --> 00:44:21.760 minus box of h. 00:44:21.760 --> 00:44:24.670 So we now have all the pieces we need 00:44:24.670 --> 00:44:27.258 to make the Einstein tensor. 00:44:27.258 --> 00:44:28.800 So I'm going to write out the result. 00:44:28.800 --> 00:44:31.383 And then we're going to stop and just look at it for a second. 00:44:48.440 --> 00:44:56.360 Einstein is Ricci minus 1/2 metric Ricci scalar. 00:44:56.360 --> 00:44:58.790 Keeping things to leading order in h. 00:45:01.694 --> 00:45:05.060 This becomes flat spacetime metric going into there. 00:45:08.170 --> 00:45:10.800 So when you put all these ingredients together, 00:45:10.800 --> 00:45:14.550 there's an overall prefactor of 1/2. 00:45:14.550 --> 00:45:18.143 And then there are 1, 2, 3, 4, 5, 6 terms. 00:45:18.143 --> 00:45:19.060 Let me write them out. 00:45:51.980 --> 00:45:52.480 OK. 00:45:57.010 --> 00:46:00.020 So recall at the beginning of the lecture, 00:46:00.020 --> 00:46:04.030 I pointed out that when one regards G alpha beta 00:46:04.030 --> 00:46:06.980 as just a differential operator on the spacetime metric, 00:46:06.980 --> 00:46:09.130 it's kind of a mess. 00:46:09.130 --> 00:46:12.370 Bearing in mind that what I have here is a simplified 00:46:12.370 --> 00:46:14.120 version of that, I have discarded 00:46:14.120 --> 00:46:18.410 all of the terms that are higher order in h than linear. 00:46:18.410 --> 00:46:21.310 This is already pretty much a bloody mess as it is. 00:46:21.310 --> 00:46:23.200 So you can sort of see my point there. 00:46:23.200 --> 00:46:25.570 If this were done in its full generality, 00:46:25.570 --> 00:46:27.145 it would be kind of a disaster. 00:46:30.630 --> 00:46:36.930 Now, in linearized theory, there is a bit of sleight of hand 00:46:36.930 --> 00:46:39.880 that lets us clean this up a little bit. 00:46:39.880 --> 00:46:42.810 Let me emphasize that the next few lines of calculation I'm 00:46:42.810 --> 00:46:45.450 going to write down, there's nothing profound. 00:46:45.450 --> 00:46:46.920 All I'm going to do is show a way 00:46:46.920 --> 00:46:49.620 of reorganizing the terms, which simplifies 00:46:49.620 --> 00:46:51.130 this in an important way. 00:46:55.200 --> 00:46:58.670 So what we're going to do is define the following tensor. 00:46:58.670 --> 00:47:10.390 h bar is h minus 1/2 eta alpha beta h. 00:47:10.390 --> 00:47:13.450 So this is a good point to go, well, who ordered that? 00:47:13.450 --> 00:47:16.120 Let's take the trace of this. 00:47:16.120 --> 00:47:19.420 Let's define h bar with no indices 00:47:19.420 --> 00:47:22.840 is what I get when I trace on this. 00:47:22.840 --> 00:47:25.793 That's going to be the trace of h. 00:47:25.793 --> 00:47:27.460 This would be the trace of h alpha beta, 00:47:27.460 --> 00:47:35.200 so I just get h back, minus 1/2 h times the trace 00:47:35.200 --> 00:47:36.620 of eta alpha beta. 00:47:36.620 --> 00:47:38.260 And the trace of eta alpha beta. 00:47:38.260 --> 00:47:41.510 This is what I get when I raise one index, 00:47:41.510 --> 00:47:43.180 and sum over the diagonal. 00:47:43.180 --> 00:47:43.990 That is 4. 00:47:47.870 --> 00:47:51.620 So the trace of h bar is negative the trace of h. 00:47:57.150 --> 00:48:01.110 We call h bar alpha beta the trace reversed 00:48:01.110 --> 00:48:02.610 metric perturbation. 00:48:02.610 --> 00:48:05.070 It's got exactly the same information 00:48:05.070 --> 00:48:06.970 as my original metric perturbation, 00:48:06.970 --> 00:48:10.800 but I've just redefined a couple terms in order 00:48:10.800 --> 00:48:16.230 to give it a trace that has the opposite sign 00:48:16.230 --> 00:48:19.170 of the original perturbation h. 00:48:19.170 --> 00:48:22.020 The reason why this is useful is recall 00:48:22.020 --> 00:48:26.550 the Einstein tensor is itself the trace reversed Ricci 00:48:26.550 --> 00:48:28.200 tensor. 00:48:28.200 --> 00:48:30.540 What we're going to see is that if we-- 00:48:30.540 --> 00:48:33.870 in acknowledgment that it's sort of a trace reverse thing, 00:48:33.870 --> 00:48:36.900 if I plug in a trace reverse metric perturbation, 00:48:36.900 --> 00:48:39.520 a couple of terms are going to get cleaned up. 00:48:39.520 --> 00:48:40.600 So here's how we do this. 00:48:40.600 --> 00:48:46.560 So let's now insert h bar. 00:48:46.560 --> 00:48:48.820 This guy is going to be equal to-- 00:48:48.820 --> 00:48:51.190 oops, pardon me. 00:48:51.190 --> 00:48:53.970 Insert h. 00:48:53.970 --> 00:49:03.350 This guy is h bar plus 1/2 eta alpha beta h. 00:49:03.350 --> 00:49:05.360 So just move that to the other side. 00:49:05.360 --> 00:49:07.160 All I'm doing is taking the definition, 00:49:07.160 --> 00:49:10.070 and I am moving part of it to the other side, 00:49:10.070 --> 00:49:13.210 so that I can substitute in for h. 00:49:13.210 --> 00:49:14.830 When you plug this into here, you'll 00:49:14.830 --> 00:49:16.780 see that there are certain cancellations. 00:49:16.780 --> 00:49:21.130 In particular, every term that involves the trace of h, 00:49:21.130 --> 00:49:23.920 h without any indices, is canceled out. 00:49:38.870 --> 00:49:42.710 And so what you find doing this algebra is that your Einstein 00:49:42.710 --> 00:49:47.595 tensor turns into-- 00:49:58.607 --> 00:49:59.440 that can't be right. 00:50:21.490 --> 00:50:22.580 OK. 00:50:22.580 --> 00:50:25.700 So now, my Einstein tensor has no trace of h in it. 00:50:25.700 --> 00:50:28.760 Every h that appears on its right hand side 00:50:28.760 --> 00:50:30.890 is the tensor with both of the indices. 00:50:30.890 --> 00:50:35.020 But now, it's the trace reverse version of that. 00:50:35.020 --> 00:50:40.000 This is still a bit of a mess. 00:50:40.000 --> 00:50:44.810 Now, we're going to do something that's got a little bit more-- 00:50:44.810 --> 00:50:46.230 it's not just sleight of hand. 00:50:46.230 --> 00:50:48.340 This is something that's got a little bit more 00:50:48.340 --> 00:50:50.650 of sort of the meaning of some of these manipulations 00:50:50.650 --> 00:50:51.567 that we've worked out. 00:50:51.567 --> 00:50:55.540 It's going to play a role in helping us to understand this. 00:50:55.540 --> 00:51:02.250 Notice this term involves delta mu on h mu. 00:51:02.250 --> 00:51:07.380 Excuse me, partial mu on h mu, partial mu on h mu. 00:51:07.380 --> 00:51:11.540 Partial mu and partial nu on h mu. 00:51:11.540 --> 00:51:16.940 This is the only term that does not look like a divergence. 00:51:28.100 --> 00:51:29.920 Three of the terms in my Einstein tensor 00:51:29.920 --> 00:51:39.660 look like divergences of the trace reverse metric. 00:51:44.650 --> 00:51:47.845 Wouldn't it be nice if we could eliminate them somehow? 00:51:55.070 --> 00:51:57.140 Well, if you studied gauge transformations 00:51:57.140 --> 00:51:59.120 and electrodynamics, you'll note that there's 00:51:59.120 --> 00:52:01.470 something similar that is done. 00:52:01.470 --> 00:52:05.690 You can choose a gauge, such the divergence of the vector 00:52:05.690 --> 00:52:06.680 for potential vanishes. 00:52:12.170 --> 00:52:21.640 Can we set the divergence of this guy equal to zero? 00:52:21.640 --> 00:52:25.660 So if you look at this, mu is a dummy index. 00:52:25.660 --> 00:52:29.230 This is four conditions that we are trying to set. 00:52:29.230 --> 00:52:31.425 This has to happen for mu-- 00:52:31.425 --> 00:52:32.800 well, we're going to sum over mu. 00:52:32.800 --> 00:52:33.300 Pardon me. 00:52:33.300 --> 00:52:35.260 It's going to happen for nu equal time, 00:52:35.260 --> 00:52:38.710 and for my three spaces. 00:52:38.710 --> 00:52:40.503 These are four conditions. 00:52:48.240 --> 00:53:03.050 My gauge generators, my xi nu are four free functions. 00:53:12.420 --> 00:53:15.120 That suggests that the gauge generators give me 00:53:15.120 --> 00:53:19.410 enough freedom that I can adjust my gauge such 00:53:19.410 --> 00:53:22.950 that if I start out with some original, 00:53:22.950 --> 00:53:27.300 I have an h old that is not divergence free. 00:53:27.300 --> 00:53:29.430 Perhaps I can make an h new that is. 00:53:32.308 --> 00:53:33.100 Well, let's try it. 00:53:55.910 --> 00:53:58.250 So remember, I just erased it. 00:53:58.250 --> 00:54:00.470 But in fact, I'll just write it down right now. 00:54:00.470 --> 00:54:03.440 The shift to the metric perturbation arising 00:54:03.440 --> 00:54:05.990 from the gauge transformation, it's on h. 00:54:05.990 --> 00:54:07.520 We need to look at how it affects 00:54:07.520 --> 00:54:08.660 the trace reverse stage. 00:54:12.400 --> 00:54:16.320 So if I start with my new perturbation 00:54:16.320 --> 00:54:21.670 is related to my old perturbation as follows. 00:54:27.270 --> 00:54:36.760 It's not too hard to show that your trace reversed 00:54:36.760 --> 00:54:38.276 metric perturbation. 00:54:45.092 --> 00:54:46.050 Pardon, pardon, pardon. 00:54:49.200 --> 00:54:57.170 My trace reverse perturbation transforms 00:54:57.170 --> 00:54:59.235 in almost the exact same way. 00:54:59.235 --> 00:55:00.110 I get one extra term. 00:55:08.940 --> 00:55:11.660 So now, what I want to do is look 00:55:11.660 --> 00:55:15.710 at how the divergence of this transforms. 00:55:29.770 --> 00:55:32.980 So I'm going to get one term here, d mu of this. 00:55:32.980 --> 00:55:39.917 It gives me a wave operator acting on my gauge generator. 00:55:42.780 --> 00:55:53.022 And then I get another term here that looks like-- 00:55:53.022 --> 00:55:54.605 remember, partial derivatives commute. 00:55:54.605 --> 00:56:12.210 So you can think of this as d nu of the divergence of xi, eta mu 00:56:12.210 --> 00:56:20.090 nu acting on this changes this into d 00:56:20.090 --> 00:56:22.670 nu on the convergence of xi. 00:56:22.670 --> 00:56:26.220 And I messed up the sign, my apologies. 00:56:26.220 --> 00:56:29.330 That plus sign should have come down here. 00:56:29.330 --> 00:56:32.000 These are equal but opposite. 00:56:32.000 --> 00:56:32.570 They cancel. 00:56:35.850 --> 00:57:03.250 So let me just highlight the result. 00:57:03.250 --> 00:57:10.760 So what this tells me is if I choose my gauge generators just 00:57:10.760 --> 00:57:18.570 right, I can adjust my trace reverse metric, 00:57:18.570 --> 00:57:20.430 so that it is divergence free. 00:57:51.910 --> 00:57:58.970 If I do that, then the first three terms in my Einstein 00:57:58.970 --> 00:58:00.410 tensor here vanish. 00:58:11.320 --> 00:58:15.420 And if I do that, then here is my Einstein tensor. 00:58:23.790 --> 00:58:30.640 So just as in e and m, all that you need to do is say, 00:58:30.640 --> 00:58:32.650 I'm going to change my gauge such 00:58:32.650 --> 00:58:35.045 that the following condition holds. 00:58:38.530 --> 00:58:41.440 The condition that describes going 00:58:41.440 --> 00:58:43.900 into this gauge such that the divergence 00:58:43.900 --> 00:58:48.070 of your trace reverse perturbation vanishes. 00:58:48.070 --> 00:58:50.170 This is a simple wave equation. 00:58:50.170 --> 00:58:53.185 So solutions to this are guaranteed to exist. 00:59:16.783 --> 00:59:18.200 If you sit down and you ask, can I 00:59:18.200 --> 00:59:20.600 come up with some kind of a pathological spacetime, 00:59:20.600 --> 00:59:22.020 or a pathological-- 00:59:22.020 --> 00:59:22.870 no. 00:59:22.870 --> 00:59:24.530 Imagine I'm in some original spacetime 00:59:24.530 --> 00:59:27.992 sufficiently pathological that doesn't allow me to do this. 00:59:27.992 --> 00:59:29.450 If you do that, you're going end up 00:59:29.450 --> 00:59:32.450 violating the conditions that define weak spacetime. 00:59:32.450 --> 00:59:34.970 You can't do that in linearized gravity anyway. 00:59:34.970 --> 00:59:38.510 So in practice, we can always choose the gauge 00:59:38.510 --> 00:59:41.600 that puts in linearized gravity my Einstein 00:59:41.600 --> 00:59:44.490 tensor in this form. 00:59:44.490 --> 00:59:48.740 This form is exactly analogous to the Lorentz gauge 00:59:48.740 --> 00:59:51.580 condition that is used in electrodynamics. 00:59:51.580 --> 00:59:58.330 And so we call this Lorentz gauge in linearized gravity. 01:00:08.220 --> 01:00:18.940 Once we've done that, here's what my Einstein field 01:00:18.940 --> 01:00:19.890 equations turn into. 01:01:03.600 --> 01:01:10.325 In the next lecture, we will solve this exactly. 01:01:13.020 --> 01:01:15.920 See, what you want to emphasize is 01:01:15.920 --> 01:01:21.600 this is one of those situations where the answer is 01:01:21.600 --> 01:01:26.310 so easy and simple for us all to work out, 01:01:26.310 --> 01:01:29.710 we don't actually really need to even do that much calculation. 01:01:29.710 --> 01:01:38.190 I'll remind you that in electrodynamics, 01:01:38.190 --> 01:01:47.160 if you work in Lorentz gauge of electrodynamics, the wave 01:01:47.160 --> 01:01:59.020 equation that governs the electromagnetic potential 01:01:59.020 --> 01:02:00.828 turns out to be-- 01:02:00.828 --> 01:02:02.620 could be factors of c and things like that, 01:02:02.620 --> 01:02:04.810 depending on which units you're working in. 01:02:04.810 --> 01:02:07.720 But we find an equation that has exactly 01:02:07.720 --> 01:02:10.150 the same mathematical structure. 01:02:10.150 --> 01:02:11.900 Possibly there's a plus sign. 01:02:11.900 --> 01:02:14.530 I should have looked that up. 01:02:14.530 --> 01:02:18.620 Wave operator on my vector potential is a source. 01:02:18.620 --> 01:02:20.658 And this is very easily solved using 01:02:20.658 --> 01:02:22.450 what's called a radiative Green's function. 01:02:38.380 --> 01:02:40.670 I will discuss this in the next lecture. 01:02:40.670 --> 01:02:44.690 You can look up the details in any advanced electrodynamics 01:02:44.690 --> 01:02:45.190 textbook. 01:02:45.190 --> 01:02:48.640 Jackson has very nice discussion of this. 01:02:48.640 --> 01:02:50.230 I have an extra index. 01:02:50.230 --> 01:02:51.790 I have a different coefficient. 01:02:51.790 --> 01:02:56.240 But the mathematical structure is identical. 01:02:56.240 --> 01:02:58.700 So as far as linearized theory is concerned, 01:02:58.700 --> 01:03:01.270 we're basically done. 01:03:01.270 --> 01:03:03.580 So I'm going to talk about the exact solution of this 01:03:03.580 --> 01:03:05.740 in the next lecture. 01:03:05.740 --> 01:03:09.170 To wrap up today's lecture, to wrap up this current lecture. 01:03:09.170 --> 01:03:13.420 Let me look at the solution of this in a particular limit. 01:03:35.130 --> 01:03:41.560 So I'm going to take my source to be 01:03:41.560 --> 01:03:50.430 a static, non-relativistic, perfect fluid. 01:03:58.450 --> 01:04:02.800 The fact that it is static means that all of my time derivatives 01:04:02.800 --> 01:04:03.400 will be zero. 01:04:08.245 --> 01:04:09.620 And if that's true for my source, 01:04:09.620 --> 01:04:12.320 it has to be true for the field that arises from it. 01:04:12.320 --> 01:04:23.300 Non-relativistic tells me that the fluid density greatly 01:04:23.300 --> 01:04:26.160 exceeds its pressure. 01:04:26.160 --> 01:04:30.000 And as a consequence, I can write my stress energy 01:04:30.000 --> 01:04:38.990 tensor as approximately density four velocity four velocity. 01:04:38.990 --> 01:04:41.690 And when you go and you look at the magnitude of these things, 01:04:41.690 --> 01:04:42.830 I sort of looked at this a little bit 01:04:42.830 --> 01:04:43.830 in the previous lecture. 01:04:46.530 --> 01:04:50.828 T00 is approximately rho. 01:04:56.020 --> 01:04:59.560 All others will be negligible. 01:04:59.560 --> 01:05:01.780 Probably there's a small correction to this, 01:05:01.780 --> 01:05:04.090 but we can neglect that on a first pass. 01:05:28.110 --> 01:05:35.970 So my field equation is dominated by the zero zero 01:05:35.970 --> 01:05:36.570 component. 01:05:43.460 --> 01:05:46.640 That's going to be the most important piece of this. 01:05:46.640 --> 01:05:49.310 Since this is static, I can immediately say-- 01:05:53.150 --> 01:05:57.559 I can change that wave operator into the plus operator. 01:06:02.610 --> 01:06:07.020 And we now notice this is exactly the equation 01:06:07.020 --> 01:06:10.610 governing the Newtonian gravitational potential, 01:06:10.610 --> 01:06:11.880 modulo a factor of four. 01:06:25.020 --> 01:06:27.250 Pardon me, factor of minus four. 01:06:27.250 --> 01:06:32.360 And so we see from this that h bar zero zero 01:06:32.360 --> 01:06:35.700 is just negative 4 times the Newtonian gravitational 01:06:35.700 --> 01:06:36.659 potential. 01:06:44.810 --> 01:06:47.150 At this order in the calculation, 01:06:47.150 --> 01:06:51.620 all other contributions to the trace reverse metric are zero. 01:07:04.300 --> 01:07:07.510 OK, so let's go from the trace reverse metric back 01:07:07.510 --> 01:07:08.110 to the metric. 01:07:11.200 --> 01:07:19.050 We use the fact that h mu nu is the trace reversed h mu nu. 01:07:19.050 --> 01:07:22.493 If we trace reverse it, we'll get the original metric back. 01:07:22.493 --> 01:07:23.910 Basically, we trace reverse twice. 01:07:31.120 --> 01:07:32.440 So the trace of this guy. 01:07:46.900 --> 01:07:50.000 OK, and so putting all these ingredients together, 01:07:50.000 --> 01:07:54.700 what we see are that the only non-zero contributions here 01:07:54.700 --> 01:07:55.753 are 8 0 0. 01:07:55.753 --> 01:07:56.920 Let's do this one carefully. 01:07:56.920 --> 01:08:13.360 This is minus 4 times Newtonian potential minus 1/2 times 8 0 0 01:08:13.360 --> 01:08:15.070 and 4 times Newtonian potential. 01:08:15.070 --> 01:08:17.060 I have a c of minus sign here. 01:08:17.060 --> 01:08:22.630 This turns into minus 2 phi n. 01:08:22.630 --> 01:08:26.479 And h1 1 equals h2 2. 01:08:26.479 --> 01:08:29.060 h equals h3 3. 01:08:29.060 --> 01:08:37.290 This is going to be 0 minus 1/2 times 1 times 4 phi n. 01:08:47.870 --> 01:08:49.130 We put all these together. 01:09:19.250 --> 01:09:31.755 And what we get is-- 01:09:54.197 --> 01:09:55.030 And I'll remind you. 01:09:55.030 --> 01:09:58.990 This is a metric that I quoted in a previous lecture 01:09:58.990 --> 01:10:01.530 that I said we would prove in an upcoming one. 01:10:01.530 --> 01:10:02.860 Well, here it is. 01:10:02.860 --> 01:10:07.078 This is the Newtonian limit of general relativity. 01:10:12.230 --> 01:10:17.360 And it's worth remarking that this thing-- 01:10:17.360 --> 01:10:20.030 we are now in the very first lecture 01:10:20.030 --> 01:10:24.250 after having derived the Einstein field equations. 01:10:24.250 --> 01:10:28.940 20 years ago, almost all laboratory tests, laboratory 01:10:28.940 --> 01:10:31.700 and astronomical observational tests of general relativity 01:10:31.700 --> 01:10:34.880 essentially came from this based on. 01:10:34.880 --> 01:10:36.770 This ends up being the foundation 01:10:36.770 --> 01:10:38.390 of gravitational lensing. 01:10:38.390 --> 01:10:41.780 This is used to look at post-Newtonian corrections 01:10:41.780 --> 01:10:44.090 in our solar system. 01:10:44.090 --> 01:10:47.270 To a good approximation, it describes a tremendous number 01:10:47.270 --> 01:10:51.770 of binary systems that we see in our galaxy 01:10:51.770 --> 01:10:53.510 and in a few other galaxies. 01:10:53.510 --> 01:10:57.650 You really need to look for a much more extreme systems 01:10:57.650 --> 01:11:02.120 before the way in which the analysis changes 01:11:02.120 --> 01:11:07.430 due to going beyond linear order starts to become important. 01:11:07.430 --> 01:11:10.850 There is an upcoming homework exercise. 01:11:10.850 --> 01:11:17.398 And for students taking this course in spring of 2020, 01:11:17.398 --> 01:11:18.815 it remains to be determined how we 01:11:18.815 --> 01:11:20.750 are going to do problem sets at this point. 01:11:20.750 --> 01:11:23.330 I will be making a decision on that in coming days. 01:11:23.330 --> 01:11:26.870 But I want to tell you about an exercise on P 01:11:26.870 --> 01:11:33.000 set number 7, in which you do a variation of this calculation. 01:11:33.000 --> 01:11:39.200 So instead of just having a static body 01:11:39.200 --> 01:11:43.930 with a body who has massive density rho, 01:11:43.930 --> 01:11:45.360 consider a rotating body. 01:11:54.253 --> 01:11:55.670 And the thing which is interesting 01:11:55.670 --> 01:11:59.240 here is that in general relativity, 01:11:59.240 --> 01:12:04.580 all forms, all fluxes of energy and momentum 01:12:04.580 --> 01:12:07.280 contribute to gravity through the stress energy tensor. 01:12:07.280 --> 01:12:10.730 So if I have a body that is rotating about an axis, 01:12:10.730 --> 01:12:11.870 there's a mass flow. 01:12:11.870 --> 01:12:14.420 There are mass currents that arise. 01:12:14.420 --> 01:12:17.030 And what you find if you do this calculation correctly 01:12:17.030 --> 01:12:19.280 is that there is a correction to the spacetime that 01:12:19.280 --> 01:12:25.680 enters into here, which reflects the fact that a rotating 01:12:25.680 --> 01:12:27.990 body generates a unique contribution 01:12:27.990 --> 01:12:32.460 to the gravity that is manifested in this space time. 01:12:32.460 --> 01:12:34.860 Now, when one looks at the behavior 01:12:34.860 --> 01:12:36.510 of a body in a spacetime like the one 01:12:36.510 --> 01:12:38.550 I've written down right here. 01:12:38.550 --> 01:12:40.645 It's very reminiscent of-- 01:12:40.645 --> 01:12:42.270 well, it's just like Newtonian gravity. 01:12:42.270 --> 01:12:43.830 It's the Newtonian limit. 01:12:43.830 --> 01:12:46.740 And Newtonian gravity looks a lot like the Coulomb 01:12:46.740 --> 01:12:48.420 electric attraction. 01:12:48.420 --> 01:12:51.820 So this is often called a gravito electric field. 01:12:51.820 --> 01:12:53.445 People use that term, particularly when 01:12:53.445 --> 01:12:57.000 they're talking about linearized general relativity. 01:12:57.000 --> 01:13:00.180 If I have a rotating body, I now have mass currents 01:13:00.180 --> 01:13:02.280 flowing in this thing. 01:13:02.280 --> 01:13:04.530 And the correction to the spacetime 01:13:04.530 --> 01:13:07.900 that arises from this, it's qualitatively quite different 01:13:07.900 --> 01:13:08.400 from us. 01:13:08.400 --> 01:13:11.850 It doesn't have that simple gravito electric Coulombic type 01:13:11.850 --> 01:13:12.960 of form. 01:13:12.960 --> 01:13:16.480 It, in fact, looks a lot more like a magnetic field. 01:13:16.480 --> 01:13:18.570 And in fact, when you ask, how does 01:13:18.570 --> 01:13:22.500 this new term that is generated affect the motion of bodies? 01:13:22.500 --> 01:13:25.170 You find something that looks a lot like the magnetic Lorentz 01:13:25.170 --> 01:13:28.900 force law describing its motion. 01:13:28.900 --> 01:13:33.210 So this is a very, very powerful tool. 01:13:33.210 --> 01:13:34.620 But it's already not enough. 01:13:34.620 --> 01:13:37.770 So we can go a lot further than this. 01:13:37.770 --> 01:13:41.700 We have done so far, the simplest possible thing 01:13:41.700 --> 01:13:46.270 that we can do with this toolkit that we have derived so far. 01:13:46.270 --> 01:13:48.270 In the next lecture that I will record, 01:13:48.270 --> 01:13:51.570 we're going to return to my linearized Einstein field 01:13:51.570 --> 01:13:53.400 equations. 01:13:53.400 --> 01:13:57.660 And I am going to explore general solutions of this. 01:13:57.660 --> 01:14:03.540 This is going to lead us into a discussion of how 01:14:03.540 --> 01:14:07.060 things behave when my gravitational source is 01:14:07.060 --> 01:14:07.560 dynamic. 01:14:10.120 --> 01:14:13.320 I do not want to lose the time derivatives that are 01:14:13.320 --> 01:14:15.150 present in that wave operator. 01:14:15.150 --> 01:14:17.400 And so this is going to lead us quite naturally, then, 01:14:17.400 --> 01:14:20.550 to a discussion of gravitational radiation. 01:14:20.550 --> 01:14:22.480 And so after the next lecture, we'll 01:14:22.480 --> 01:14:25.350 spend a lecture or two discussing the nature 01:14:25.350 --> 01:14:26.910 of gravitational radiation. 01:14:26.910 --> 01:14:29.340 And there will be an upcoming homework assignment 01:14:29.340 --> 01:14:32.340 or two in which you explore the properties 01:14:32.340 --> 01:14:34.790 of gravitational radiation.